Based on the SYZ conjecture, it is a folklore that "mirror of Lagrangian multi-sections are locally free sheaves". However, due to the non-trivial contribution of holomorphic disks and wall-crossing phenomenon, this folklore has only been verified in very few cases. In this talk, I would like to introduce the notion of tropical Lagrangian multi-sections over any integral affine manifold with singularities $$extract_itex$$B$$/extract_itex$$ equipped with a polyhedral decomposition $$extract_itex$$mathscr{P}$$/extract_itex$$. As the name suggests, tropical Lagrangian multi-sections are tropical/combinatorial replacement of Lagrangian multi-sections in the SYZ program. Given a tropical Lagrangian multi-section, together with some linear algebra data, we are going to construct a locally free sheaf $$extract_itex$$mathcal{E}_0(mathbb{L})$$/extract_itex$$ on the log Calabi-Yau space $$extract_itex$$X_0(B,mathscr{P})$$/extract_itex$$ associated to $$extract_itex$$(B,mathscr{P})$$/extract_itex$$, which plays a key role in the famous Gross-Siebert program. The locally free sheaves arise as such construction are not arbitrary and we call them tropical locally free sheaves. I will also provide the reverse construction and establish a correspondence between isomorphism classes of tropical locally free sheaves and tropical Lagrangian multi-sections modulo certain non-trivial equivalence. I will also provide a combinatorial criterion for the smoothability of the pair $$extract_itex$$(X_0(B,mathscr{P}),mathcal{E}_0(mathbb{L}))$$/extract_itex$$ in dimension 2 under some additional assumptions on $$extract_itex$$mathscr{P}$$/extract_itex$$ and $$extract_itex$$mathbb{L}$$/extract_itex$$. Finally, if time permits, I will discuss how to prove homological mirror symmetry between Lagrangian multi-sections and locally free shaves using microlocal sheaf theory.